Transcript

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Game theory
From Wikipedia, the free encyclopedia
Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology,
engineering, political science, international relations, computer science, and philosophy. Game theory attempts to mathematically capture
behavior in strategic situations, in which an individual's success in making choices depends on the choices of others. While initially
developed to analyze competitions in which one individual does better at another's expense (zero sum games), it has been expanded to
treat a wide class of interactions, which are classified according to several criteria. Today, "game theory is a sort of umbrella or 'unified field'
theory for the rational side of social science, where 'social' is interpreted broadly, to include human as well as non-human players
(computers, animals, plants)" (Aumann 1987).
Traditional applications of game theory attempt to find equilibria in these games. In an equilibrium, each player of the game has adopted a
strategy that they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an
attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often
overlap or coincide. This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium
concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.
Although some developments occurred before it, the field of game theory came into being with the 1944 book Theory of Games and
Economic Behavior by John von Neumann and Oskar Morgenstern. This theory was developed extensively in the 1950s by many scholars.
Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game
theory has been widely recognized as an important tool in many fields. Eight game theorists have won Nobel prizes in economics, and John
Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.
Contents
[hide]
• 1 Representation of games
o 1.1 Extensive form
o 1.2 Normal form
o 1.3 Characteristic function form
o 1.4 Partition function form
• 2 Application and challenges
o 2.1 Political science
o 2.2 Economics and business
 2.2.1 Descriptive
 2.2.2 Prescriptive or normative analysis
o 2.3 Biology
o 2.4 Computer science and logic
o 2.5 Philosophy
• 3 Types of games
o 3.1 Cooperative or non-cooperative
o 3.2 Symmetric and asymmetric
o 3.3 Zero-sum and non-zero-sum
o 3.4 Simultaneous and sequential
o 3.5 Perfect information and imperfect information
o 3.6 Infinitely long games
o 3.7 Discrete and continuous games
o 3.8 One-player and many-player games

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o 3.9 Metagames
• 4 History
• 5 See also
• 6 Notes
• 7 References
o 7.1 Textbooks and general references
o 7.2 Historically important texts
o 7.3 Other print references
o 7.4 Websites
Representation of games
See also: List of games in game theory
The games studied in game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies)
available to those players, and a specification of payoffs for each combination of strategies. Most cooperative games are presented in the
characteristic function form, while the extensive and the normal forms are used to define noncooperative games.
Extensive form
Main article: Extensive form game
An extensive form game
The extensive form can be used to formalize games with some important order. Games here are often presented as trees (as pictured to the
left). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines
out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree.
In the game pictured here, there are two players. Player 1 moves first and chooses either F or U. Player 2 sees Player 1's move and then
chooses A or R. Suppose that Player 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2 gets 2.
The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line
connects different vertices to represent them as being part of the same information set (i.e., the players do not know at which point they
are), or a closed line is drawn around them.
Player 2 Player 2
chooses Left chooses Right
Normal form
Player 1 4, 3 –1, –1
chooses Up
Main article: Normal-form game
Player 1 0, 0 3, 4
chooses Down
The normal (or strategic form) game is usually represented by a matrix
Normal form or payoff matrix of a 2-player, 2-strategy
which shows the players, strategies, and payoffs (see the example to the
game
right). More generally it can be represented by any function that associates
a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the
row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of
columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the

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second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then
Player 1 gets a payoff of 4, and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of
the other. If players have some information about the choices of other players, the game is usually presented in extensive form.
Characteristic function form
Main article: Cooperative game
In cooperative games with transferable utility no individual payoffs are given. Instead, the characteristic function determines the payoff of
each coalition. The standard assumption is that the empty coalition obtains a payoff of 0.
The origin of this form is to be found in the seminal book of von Neumann and Morgenstern who, when studying coalitional normal form
games, assumed that when a coalition C forms, it plays against the complementary coalition ( ) as if they were playing a 2-player
game. The equilibrium payoff of C is characteristic. Now there are different models to derive coalitional values from normal form games, but
not all games in characteristic function form can be derived from normal form games.
Formally, a characteristic function form game (also known as a TU-game) is given as a pair (N,v), where N denotes a set of players and
is a characteristic function.
The characteristic function form has been generalised to games without the assumption of transferable utility.
Partition function form
The characteristic function form ignores the possible externalities of coalition formation. In the partition function form the payoff of a coalition
depends not only on its members, but also on the way the rest of the players are partitioned (Thrall & Lucas 1963).
Application and challenges
Game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a
large collection of economic behaviors, including behaviors of firms, markets, and consumers. The use of game theory in the social sciences
has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.
Game theoretic analysis was initially used to study animal behavior by Ronald Fisher in the 1930s (although even Charles Darwin makes a
few informal game theoretic statements). This work predates the name "game theory", but it shares many important features with this field.
The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of
Games.
In addition to being used to predict and explain behavior, game theory has also been used to attempt to develop theories of ethical or
normative behavior. In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper
behavior. Game theoretic arguments of this type can be found as far back as Plato.[1]
Political science
The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice,
positive political theory, and social choice theory. In each of these areas, researchers have developed game theoretic models in which the
players are often voters, states, special interest groups, and politicians.

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For early examples of game theory applied to political science, see the work of Anthony Downs. In his book An Economic Theory of
Democracy (Downs 1957), he applies a hotelling firm location model to the political process. In the Downsian model, political candidates
commit to ideologies on a one-dimensional policy space. The theorist shows how the political candidates will converge to the ideology
preferred by the median voter. For more recent examples, see the books by Steven Brams, George Tsebelis, Gene M. Grossman and
Elhanan Helpman, or David Austen-Smith and Jeffrey S. Banks.
A game-theoretic explanation for democratic peace is that public and open debate in democracies send clear and reliable information
regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions
will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a
dispute is a nondemocracy (Levy & Razin 2003).
Economics and business
Economists have long used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, fair
division, oligopolies, social network formation, and voting systems. This research usually focuses on particular sets of strategies known as
equilibria in games. These "solution concepts" are usually based on what is required by norms of rationality. In non-cooperative games, the
most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other
strategies. So, if all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their
strategy is the best they can do given what others are doing.
The payoffs of the game are generally taken to represent the utility of individual players. Often in modeling situations the payoffs represent
money, which presumably corresponds to an individual's utility. This assumption, however, can be faulty.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation.
One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the
appropriate type. Naturally one might wonder to what use should this information be put. Economists and business professors suggest two
primary uses: descriptive and prescriptive.
Descriptive
A three stage Centipede Game
The first known use is to inform us about how actual human populations behave. Some scholars believe that by finding the equilibria of
games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied.
This particular view of game theory has come under recent criticism. First, it is criticized because the assumptions made by game theorists
are often violated. Game theorists may assume players always act in a way to directly maximize their wins (the Homo economicus model),
but in practice, human behavior often deviates from this model. Explanations of this phenomenon are many; irrationality, new models of
deliberation, or even different motives (like that of altruism). Game theorists respond by comparing their assumptions to those used in
physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models
used by physicists. However, additional criticism of this use of game theory has been levied because some experiments have demonstrated
that individuals do not play equilibrium strategies. For instance, in the centipede game, guess 2/3 of the average game, and the dictator
game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments.[2]
Alternatively, some authors claim that Nash equilibria do not provide predictions for human populations, but rather provide an explanation for
why populations that play Nash equilibria remain in that state. However, the question of how populations reach those points remains open.

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Some game theorists have turned to evolutionary game theory in order to resolve these worries. These models presume either no rationality
or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in
the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning
(for example, fictitious play dynamics).
Prescriptive or normative analysis
Cooperate Defect
On the other hand, some scholars see game theory not as a predictive tool for the behavior of
human beings, but as a suggestion for how people ought to behave. Since a Nash equilibrium Cooperate -1, -1 -10, 0
of a game constitutes one's best response to the actions of the other players, playing a
Defect 0, -10 -5, -5
strategy that is part of a Nash equilibrium seems appropriate. However, this use for game
theory has also come under criticism. First, in some cases it is appropriate to play a non- The Prisoner's Dilemma
equilibrium strategy if one expects others to play non-equilibrium strategies as well. For an
example, see Guess 2/3 of the average.
Second, the Prisoner's dilemma presents another potential counterexample. In the Prisoner's Dilemma, each player pursuing his own self-
interest leads both players to be worse off than had they not pursued their own self-interests.
Biology
Hawk Dove
Unlike economics, the payoffs for games in biology are often interpreted as corresponding to
Hawk v−c, v−c 2v, 0
fitness. In addition, the focus has been less on equilibria that correspond to a notion of
rationality, but rather on ones that would be maintained by evolutionary forces. The best known Dove 0, 2v v, v
equilibrium in biology is known as the evolutionarily stable strategy (or ESS), and was first
The hawk-dove game
introduced in (Smith & Price 1973). Although its initial motivation did not involve any of the
mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.
In biology, game theory has been used to understand many different phenomena. It was first used to explain the evolution (and stability) of
the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who
could be seen as trying to maximize their number of grandchildren.
Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication (Harper &
Maynard Smith 2003). The analysis of signaling games and other communication games has provided some insight into the evolution of
communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a
larger predator, seems to be an example of spontaneous emergent organization.
Biologists have used the game of chicken to analyze fighting behavior and territoriality.[citation needed]
Maynard Smith, in the preface to Evolution and the Theory of Games, writes, "[p]aradoxically, it has turned out that game theory is more
readily applied to biology than to the field of economic behaviour for which it was originally designed". Evolutionary game theory has been
used to explain many seemingly incongruous phenomena in nature.[3]
One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other
organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear
to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate
blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the
queen bee for their entire lives and never mate, to Vervet monkeys that warn group members of a predator's approach, even when it
endangers that individual's chance of survival.[4] All of these actions increase the overall fitness of a group, but occur at a cost to the
individual.

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Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help; and
favor relatives. Hamilton's rule explains the evolutionary reasoning behind this selection with the equation c<b*r where the cost ( c ) to the
altruist must be less than the benefit ( b ) to the recipient multiplied by the coefficient of relatedness ( r ). The more closely related two
organisms are causes the incidence of altruism to increase because they share many of the same alleles. This means that the altruistic
individual, by ensuring that the alleles of its close relative are passed on, (through survival of its offspring) can forgo the option of having
offspring itself because the same number of alleles are passed on. Helping a sibling for example, has a coefficient of ½, because an
individual shares ½ of the alleles in its sibling's offspring. Ensuring that enough of a sibling’s offspring survive to adulthood precludes the
necessity of the altruistic individual producing offspring.[4] The coefficient values depend heavily on the scope of the playing field; for
example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all
humans only accounts for approximately 1% of the diversity in the playing field, a co-efficient that was ½ in the smaller field becomes 0.995.
Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc) persisted through time
the playing field becomes larger still, and the discrepancies smaller.
Computer science and logic
Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in
game semantics. In addition, computer scientists have used games to model interactive computations. Also, game theory provides a
theoretical basis to the field of multi-agent systems.
Separately, game theory has played a role in online algorithms. In particular, the k-server problem, which has in the past been referred to as
games with moving costs and request-answer games (Ben David, Borodin & Karp et al. 1994). Yao's principle is a game-theoretic technique
for proving lower bounds on the computational complexity of randomized algorithms, and especially of online algorithms.
The field of algorithmic game theory combines computer science concepts of complexity and algorithm design with game theory and
economic theory. The emergence of the internet has motivated the development of algorithms for finding equilibria in games, markets,
computational auctions, peer-to-peer systems, and security and information markets.[5]
Philosophy
Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Stag Hare
Quine (1960, 1967), Lewis (1969) used game theory to develop a philosophical account of
Stag 3, 3 0, 2
convention. In so doing, he provided the first analysis of common knowledge and employed it
in analyzing play in coordination games. In addition, he first suggested that one can Hare 2, 0 2, 2
understand meaning in terms of signaling games. This later suggestion has been pursued by
Stag hunt
several philosophers since Lewis (Skyrms (1996), Grim, Kokalis, and Alai-Tafti et al. (2004)).
Following Lewis (1969) game-theoretic account of conventions, Ullmann Margalit (1977) and Bicchieri (2006) have developed theories of
social norms that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.[6]
Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for a collective to have common
beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from agents' interactions.
Philosophers who have worked in this area include Bicchieri (1989, 1993),[7] Skyrms (1990),[8] and Stalnaker (1999).[9]
In ethics, some authors have attempted to pursue the project, begun by Thomas Hobbes, of deriving morality from self-interest. Since
games like the Prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by
self-interest is an important component of this project. This general strategy is a component of the general social contract view in political
philosophy (for examples, see Gauthier (1986) and Kavka (1986).[10]
Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and
corresponding animal behaviors. These authors look at several games including the Prisoner's dilemma, Stag hunt, and the Nash
bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., Skyrms (1996, 2004) and Sober and
Wilson (1999)).

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Some assumptions used in some parts of game theory have been challenged in philosophy; psychological egoism states that rationality
reduces to self-interest—a claim debated among philosophers. (see Psychological egoism#Criticism)
Types of games
Cooperative or non-cooperative
Main articles: Cooperative game and Non-cooperative game
A game is cooperative if the players are able to form binding commitments. For instance the legal system requires them to adhere to their
promises. In noncooperative games this is not possible.
Often it is assumed that communication among players is allowed in cooperative games, but not in noncooperative ones. This classification
on two binary criteria has been rejected (Harsanyi 1974).
Of the two types of games, noncooperative games are able to model situations to the finest details, producing accurate results. Cooperative
games focus on the game at large. Considerable efforts have been made to link the two approaches. The so-called Nash-
programme[clarification needed] has already established many of the cooperative solutions as noncooperative equilibria.
Hybrid games contain cooperative and non-cooperative elements. For instance, coalitions of players are formed in a cooperative game, but
these play in a non-cooperative fashion.
E F
Symmetric and asymmetric
E 1, 2 0, 0
Main article: Symmetric game
F 0, 0 1, 2
A symmetric game is a game where the payoffs for playing a particular strategy depend only An asymmetric game
on the other strategies employed, not on who is playing them. If the identities of the players
can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are
symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some scholars
would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these
games are symmetric.
Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the
ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical
strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy
sets for both players.
A B
Zero-sum and non-zero-sum
A –1, 1 3, –3
Main article: Zero-sum (game theory)
B 0, 0 –2, 2
Zero-sum games are a special case of constant-sum games, in which choices by players can A zero-sum game
neither increase nor decrease the available resources. In zero-sum games the total benefit to
all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal
expense of others). Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount
one's opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and chess.
Many games studied by game theorists (including the famous prisoner's dilemma) are non-zero-sum games, because some outcomes have
net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss
by another.

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Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are
potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding an additional dummy
player (often called "the board"), whose losses compensate the players' net winnings.
Simultaneous and sequential
Main article: Sequential game
Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are
unaware of the earlier players' actions (making them effectively simultaneous). Sequential game (or dynamic games) are games where later
players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be
very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while he does not know
which of the other available actions the first player actually performed.
The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal
form is used to represent simultaneous games, and extensive form is used to represent sequential ones; although this isn't a strict rule in a
technical sense.
Perfect information and imperfect information
A game of imperfect information (the dotted line represents ignorance on the part of player 2)
Main article: Perfect information
An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know
the moves previously made by all other players. Thus, only sequential games can be games of perfect information, since in simultaneous
games not every player knows the actions of the others. Most games studied in game theory are imperfect-information games, although
there are some interesting examples of perfect-information games, including the ultimatum game and centipede game. Perfect-information
games include also chess, go, mancala, and arimaa.
Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player
know the strategies and payoffs of the other players but not necessarily the actions.
Infinitely long games
Main article: Determinacy
Games, as studied by economists and real-world game players, are generally finished in a finite number of moves. Pure mathematicians are
not so constrained, and set theorists in particular study games that last for an infinite number of moves, with the winner (or other payoff) not
known until after all those moves are completed.
The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player
has a winning strategy. (It can be proven, using the axiom of choice, that there are games—even with perfect information, and where the
only outcomes are "win" or "lose"—for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed
games, has important consequences in descriptive set theory.

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Discrete and continuous games
Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc. Many
concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set. For instance,
Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.
Differential games such as the continuous pursuit and evasion game are continuous games.
One-player and many-player games
Individual decision problems are sometimes considered "one-player games". While these situations are not game theoretical, they are
modeled using many of the same tools within the discipline of decision theory. It is only with two or more players that a problem becomes
game theoretical. A randomly acting player who makes "chance moves", also known as "moves by nature", is often added (Osborne &
Rubinstein 1994). This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide
a roll of the dice where required by the game. Games with an infinite number of players are often called n-person games (Luce & Raiffa
1957).
Metagames
These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to
maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory.
History
The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a
minimax mixed strategy solution to a two-person version of the card game le Her.
James Madison made what we now recognize as a game-theoretic analysis of the ways states can be expected to behave under different
systems of taxation.[11][12]
It was not until the publication of Antoine Augustin Cournot's Recherches sur les principes mathématiques de la théorie des richesses
(Researches into the Mathematical Principles of the Theory of Wealth) in 1838 that a general game theoretic analysis was pursued. In this
work Cournot considers a duopoly and presents a solution that is a restricted version of the Nash equilibrium.
Although Cournot's analysis is more general than Waldegrave's, game theory did not really exist as a unique field until John von Neumann
published a series of papers in 1928. While the French mathematician Émile Borel did some earlier work on games, Von Neumann can
rightfully be credited as the inventor of game theory. Von Neumann was a brilliant mathematician whose work was far-reaching from set
theory to his calculations that were key to development of both the Atom and Hydrogen bombs and finally to his work developing computers.
Von Neumann's work in game theory culminated in the 1944 book Theory of Games and Economic Behavior by von Neumann and Oskar
Morgenstern. This profound work contains the method for finding mutually consistent solutions for two-person zero-sum games. During this
time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of
individuals, presuming that they can enforce agreements between them about proper strategies.
In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation.
Around this same time, John Nash developed a criterion for mutual consistency of players' strategies, known as Nash equilibrium,
applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. This equilibrium is sufficiently general
to allow for the analysis of non-cooperative games in addition to cooperative ones.
Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious
play, repeated games, and the Shapley value were developed. In addition, the first applications of Game theory to philosophy and political
science occurred during this time.

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In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium (later he
would introduce trembling hand perfection as well). In 1967, John Harsanyi developed the concepts of complete information and Bayesian
games. Nash, Selten and Harsanyi became Economics Nobel Laureates in 1994 for their contributions to economic game theory.
In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily
stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[13] were introduced
and analyzed.
In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked
on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing an
equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of
its consequences.
In 2007, Roger Myerson, together with Leonid Hurwicz and Eric Maskin, was awarded the Nobel Prize in Economics "for having laid the
foundations of mechanism design theory." Myerson's contributions include the notion of proper equilibrium, and an important graduate text:
Game Theory, Analysis of Conflict (Myerson 1997).
See also
• Combinatorial game theory
• Glossary of game theory
• List of games in game theory
• Quantum game theory
• Self-confirming equilibrium
Notes
1. ^ Ross, Don. "Game Theory". The Stanford Encyclopedia of Philosophy (Spring 2008 Edition). Edward N. Zalta (ed.).
http://plato.stanford.edu/archives/spr2008/entries/game-theory/. Retrieved 2008-08-21.
2. ^ Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioural game
theory are several. For a recent discussion on this field see Camerer (2003).
3. ^ Evolutionary Game Theory (Stanford Encyclopedia of Philosophy)
4. ^ a b Biological Altruism (Stanford Encyclopedia of Philosophy)
5. ^ Algorithmic Game Theory. http://www.cambridge.org/journals/nisan/downloads/Nisan_Non-printable.pdf.
6. ^ E. Ullmann Margalit, The Emergence of Norms, Oxford University Press, 1977. C. Bicchieri, The Grammar of Society: the Nature
and Dynamics of Social Norms, Cambridge University Press, 2006.
7. ^ "Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge ", Erkenntnis 30, 1989: 69-85. See also
Rationality and Coordination, Cambridge University Press, 1993.
8. ^ The Dynamics of Rational Deliberation, Harvard University Press, 1990.
9. ^ "Knowledge, Belief, and Counterfactual Reasoning in Games." In Cristina Bicchieri, Richard Jeffrey, and Brian Skyrms, eds., The
Logic of Strategy. New York: Oxford University Press, 1999.
10. ^ For a more detailed discussion of the use of Game Theory in ethics see the Stanford Encyclopedia of Philosophy's entry game
theory and ethics.
11. ^ James Madison, Vices of the Political System of the United States, April, 1787. Link
12. ^ Jack Rakove, "James Madison and the Constitution", History Now, Issue 13 September 2007. Link
13. ^ Although common knowledge was first discussed by the philosopher David Lewis in his dissertation (and later book) Convention
in the late 1960s, it was not widely considered by economists until Robert Aumann's work in the 1970s.